A Posteriori Estimates for Partial Differential Equations
A posteriori error estimates have attracted much attention of numerical analysts in the last few decades because they allow reliable verification of numerically obtained approximate solutions. This is possible since these estimates enable one to find lower and upper bounds of the distance of a given approximate solution of the studied partial differential equation and the unknown exact solution. These bounds are constructed directly for the given approximated solution. The book starts with a general introduction to the theory of error control. It continues with an overview of classical a posteriori estimate methods developed in the 20th century. The main part of the book deals with new functional a posteriori estimates. The author explains the method on the Poisson equation with Dirichlet boundary conditions. Then he also applies it to the Poisson equation with different boundary conditions, problems arising in linear plasticity and in the theory of viscous fluids, variational inequalities and some other problems. The book contains not only estimates for picked problems but attention is also focused on the methods of how these estimates are derived. This is important, since understanding the method allows one to modify it for different problems of interest. The text requires a moderate background in functional analysis and the theory of partial differential equations. It will be useful for experts in computational mathematics, as well as for students interested in applied mathematics.